GxE interaction
Biological interaction: Genetic factor(s) and environmental factor(s) participate in the same causal mechanism in the same individual (Rothman et al., 2008)
Statistical interaction using linear regression (unrelated individuals):
\(y = \mu + \beta_g x_g + \beta_e x_e + \beta_{int} x_g \times x_e + e\)
GxE interaction in GWAS context
| CHARGES + SPIROMETA |
50,047 |
Smoking |
Pulmonary function |
(Hancock et al., 2012) |
| SUNLIGHT |
35,000 |
Vitamin D intake |
Circulating Vitamin D level |
(Wang et al., 2010) |
| GIANT |
up to 339,224 |
Gender |
Anthropometric traits |
(Heid et al., 2010) |
| … |
|
|
|
|
Most have used the aforementioned full model (on the previous slide), but others used stratified approach (coming on the next slides).
GxE interaction in GWAS context
Example: G x smoking in pulmonary function outcomes (Hancock et al., 2012)
- 50,047 participants from 19 studies; ~2.5M SNPs
- outcomes: FEV1, FEV1/FVC (%)
- saturated model for smoking: ever-smoker, current-smoker, packs-year
- all three smoking variables tested for GxE separately
- joint test: \(\beta_g = 0\) and \(\beta_{int} = 0\) under the null (Aschard et al., 2011)
Findings: three novel gene regions
- DNER
- HLA-DQB1 and HLA-DQA2
- KCNJ2 and SOX9
GxE interaction in GWAS context
Stratified GxE tests (Magi et al., 2010), (Randall et al., 2013) are widely used in meta-analysis by big consortia
Example: G x gender in the Genetic Investigation of Anthropometric Traits (GIANT) consortium
- 14 WHR associated SNPs
- 108,979 women
- 42,735 (discovery) and 66,244 (follow up)
- 82,483 men
- 34,601 (discovery) and 47,882 (follow up)
Findings: 7 loci showed sex-specificity
- explained variance in WHR by 14 loci
- 1.34% in women
- 0.46% in men
Study design considered
| Sample |
Family-based |
Population-based |
Population-based |
| Relationships |
Kinship |
|
GRM |
| Method |
Linear mixed models |
Linear models |
Linear mixed models |
GxE in study design 3 is our ongoing work (not presented today)
GxE in study designs 1 vs. 2 (today focus)
- Compare two study designs unrelated/related
- LMM performed using our new lme4qtl R package (under submission)
Simulation study
Given: a population of 50,000 related samples (nuclear families)
Experiment: pool 5,000 unrelated samples or pool randomly
| unrelated |
\(\sigma_g^2 K + \sigma_r^2 I = (\sigma_g^2 + \sigma_r^2) I\) |
\(\sigma_x I\) |
\(\sigma_g^2 + \sigma_r^2 = 1\) |
| genetically related |
\(\sigma_g^2 K + \sigma_r^2 I\) |
\(\sigma_x K\) |
\(\sigma_g^2 + \sigma_r^2 = 1\) |
- \(K\), the double kinship matrix
- \(\Sigma_x\), the variance-covariance matrix of predictor \(x\)
- \(\sigma_g^2\), \(\sigma_r^2\), variance proportions
Linear mixed model (also our data simulation model)
\(y = \mu + \beta_g x_g + g + f + e\)
\(\mbox{ } \mbox{ } = X \beta + g + f + e\)
\(\mbox{where } g \perp f \perp e\)
- \(g \sim (0, \sigma_g^2 K)\), the (additive) genetic effect
- \(h \sim (0, \sigma_f^2 F)\), the house-hold/family effect
- \(e \sim (0, \sigma_r^2 I)\), the residual error
\(\mbox{implying}\)
\(y \sim (X \beta, \sigma_g^2 K + \sigma_f^2 F + \sigma_r^2 I) = (X \beta, V)\)
Marginal effect estimate
- Estimate variance components by ML/REML
\(\hat{V} = \hat{\sigma_g^2} K + \hat{\sigma_f^2} F + \hat{\sigma_r^2} I\)
- Derive the effect size as in Generalized Least Squares (GLS)
\(\hat{\beta}_g = (X^T \hat{V}^{-1} X)^{-1} X^T \hat{V}^{-1} Y\)
\(\hat{\beta}_g \simeq \mathcal{N}(\beta, (X^T \hat{V}^{-1} X)^{-1})\)
- Simplify to a one-covariate model by orthogonalization
- the original model: \(E(Y) = \mu + \beta_g x_g\)
- \(y^*\), centered \(y\)
- \(x^*_g\), centered \(x_g\)
\(var(\hat{\beta}_g) = ({x^*_g}^T \hat{V}^{-1} x^*_g)^{-1}\)
Power
The power in genetic association studies with linear models is a function of the non-centrality parameter (NCP)
\(NCP \approx \beta^2 tr(\hat{V}^{-1} \Sigma_x)\)
| phenotype |
\(y \sim (X \beta, V) = (X \beta, \sigma_f^2 F + \sigma_g^2 K + \sigma_r^2 I)\) |
| genotypes |
\(x \sim (\mu_x, \Sigma_x)\) |
Type I error rate
The standard error from a fixed effect LM applied in related individuals is not well calibrated (i.e. underestimated)
Compare formula
Confirmed the known results
But our formula allows us to explore further performances across various study designs
Conclusion
- GxE analysis in genetically related individuals is slightly less powerful than in unrelated
- GxE analysis that takes into account shared environment relatedness always increases the power
- more variance is explained
